8 COMPLETE HUBERT ALGEBRAS 363 In view of (15.8.6) and (15.8.3), we obtain the following corollary: (15.8.8) Every left ideal in A contains a minimal left ideal. Every minimal left ideal is closed. (15.8.9) (i) If e, e' are two orthogonal self-adjoint idempotents, then the left ideals Ae and Ae' are orthogonal. (ii) Let (e^i^i^n be a finite family of pairwise orthogonal, self-adjoint idempotents. Then for each x e A, the element x — ]T xe-t is orthogonal to each i-l (i) We have (xe\ye*) = (xe\ye'2) = (xee'\yef) = 0, by (15.7.5.5) and (15.8.4). (ii) Likewise, (n \ / n \ * - E ***Iyej] = (xej - ExeiejIyejj = ° for all/ (15.8.10) For each XE A, there exists a finite or infinite sequence (en) of pair- wise orthogonal, irreducible self-adoint idempotents belonging to the closure I of the ideal Ax, such that x = ]T xen (this series being convergent in A), and We may assume that x ^ 0. Then there exists at least one self-adjoint idempotent e € I such that xe ^ 0, for the construction in (15.8.5) produces a self-adjoint idempotent e ^ 0 in I such that e(x*x)2 = (?, and therefore ex* =£ 0, so that xe = (ex*)* ^ 0. Since e is the sum of a finite number of irreducible self-adjoint idempotents belonging to Ae (15.8.6), there exists at least one of these idempotents whose product with x is ^0. We next remark that if (ej)i $ i^n is a finite sequence of pairwise orthogonal, self-adjoint idem- potents such that \\xet\\2 ^ a > 0 for all z, then by virtue of (15.8.9) we have i=l || i=l 1=1 and therefore n gjj ||x||2/a. Now define inductively an increasing sequence (<p(«))n^0 of integers ^0; a (finite or infinite) sequence (ek) of pairwise orthogonal, irreducible self-adjoint idempotents belonging to I; and a e2(e - e2) = 0,